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S3.3StatisticsCore

Cumulative Frequency

A cumulative frequency graph plots running totals against the upper class boundaries. The resulting S-shaped curve lets you read off the median, quartiles and the number of values below any given amount.

25 min Representations of Data
Edexcel Statistics Y1 — Representations of Data playlist (Zeeshan Zamurred)Curated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Build a cumulative frequency table
  • Plot a cumulative frequency graph
  • Read off the median and quartiles
  • Estimate how many values lie below a value
1

Cumulative frequency

The cumulative frequency\textbf{cumulative frequency} at a point is the running total of all frequencies up to and including that class. Plot it against the upper class boundary\textbf{upper class boundary} of each class and join the points with a smooth curve.

Tip — Always plot cumulative frequency against the UPPER boundary of each class — a common slip is to use the midpoint.

2

Reading off statistics

From the curve, read across from the relevant cumulative frequency to find a value. For nn data points, the median is at n2\tfrac{n}{2}, Q1Q_1 at n4\tfrac{n}{4} and Q3Q_3 at 3n4\tfrac{3n}{4} on the cumulative axis.

median at n2,Q1 at n4,Q3 at 3n4\text{median at } \tfrac{n}{2}, \quad Q_1 \text{ at } \tfrac{n}{4}, \quad Q_3 \text{ at } \tfrac{3n}{4}
Read across from these heights, then down to the value axis.
3

Values below a number

To estimate how many data points are below a given value, read up from that value to the curve and across to the cumulative frequency. Subtract from nn for “how many above”.

Example 1A curve shows cumulative frequency 30 at x=50x = 50 (total 80). How many values exceed 50?
1Below 50: 30. Above: 80−3080 - 30.
Answer50 values

Formula recap

cf=running total of frequencies\text{cf} = \text{running total of frequencies}
Cumulative frequency.
plot cf vs upper class boundary\text{plot cf vs upper class boundary}
How to plot.
median n2, Q1 n4, Q3 3n4\text{median } \tfrac{n}{2}, \ Q_1 \ \tfrac{n}{4}, \ Q_3 \ \tfrac{3n}{4}
Reading quartiles.

Common mistakes to avoid

Plotting cumulative frequency against the class midpoint.
Use the UPPER class boundary.
Reading the median at (n+1)/2 on a cumulative frequency graph.
For a large grouped data curve, use n/2.

Key takeaways

  • Cumulative frequency is a running total; plot it vs upper class boundaries.
  • Read median, Q₁, Q₃ at n/2, n/4, 3n/4 on the cumulative axis.
  • Read across/down to estimate values below a given number.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.